# Mathematical maturity

Mathematical maturity often refers to the mastery of the way mathematicians think, operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structures and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right.

## Definitions

Mathematical maturity has been defined in several different ways by various authors, and is often tied to other related concepts such as comfort and competence with mathematics, mathematical intuition and mathematical beliefs.
One definition has been given as follows:
   ... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas.
A broader list of characteristics of mathematical maturity has been given as follows:
   * The capacity to generalize from a specific example to a broad concept
   * The capacity to handle increasingly abstract ideas
   * The ability to communicate mathematically by learning standard notation and acceptable style
   * A significant shift from learning by memorization to learning through understanding
   * The capacity to separate the key ideas from the less significant
   * The ability to link a geometrical representation with an analytic representation
   * The ability to translate verbal problems into mathematical problems
   * The ability to recognize a valid proof and detect 'sloppy' thinking
   * The ability to recognize mathematical patterns
   * The ability to move back and forth between the geometrical (graph) and the analytical (equation)
   * Improving mathematical intuition by abandoning naive assumptions and developing a more critical attitude
Finally, mathematical maturity has also been defined as an ability to do the following:
   * Make and use connections with other problems and other disciplines
   * Fill in missing details
   * Spot, correct and learn from mistakes
   * Winnow the chaff from the wheat, get to the crux, identify intent
   * Recognize and appreciate elegance
   * Think abstractly
   * Read, write and critique formal proofs
   * Draw a line between what you know and what you don’t know
   * Recognize patterns, themes, currents and eddies
   * Apply what you know in creative ways
   * Approximate appropriately
   * Teach yourself
   * Generalize
   * Remain focused
   * Bring instinct and intuition to bear when needed
It is sometimes said that the development of mathematical maturity requires a deep reflection on the subject matter for a prolonged period of time, along with a guiding spirit which encourages exploration.

## Progression

Mathematician Terence Tao has proposed a three-stage model of mathematics education that can be interpreted as a general framework of mathematical maturity progression. The stages are summarized in the following table:
Overview of Terence Tao's stages of mathematical understanding :
| Stage | Level | Phase | Description | Typical duration | Nature of mistakes | Example |
| 1 | Beginner | Pre-rigorous | This stage occurs before a mathematics student has learned mathematical formalism. Mathematics is learned in an informal manner that often incorporates examples, fuzzy notions and hand-waving. This stage is founded mostly in intuition and the emphasis is more on computation than theory. Natural talent is important at this level. | Until underclassman undergraduate years | Formal errors are made as a result of the application of formal rules or heuristics blindly due to the misunderstanding of rigorous mathematical formalism. Such errors can be difficult to correct even when pointed out. | The introduction of calculus in terms of slopes, areas, rates of change etc. |
| 2 | Intermediate | Rigorous | Students learn mathematical formalism in this phase, which typically includes formal logic and some mathematical logic. This serves a foundation for learning mathematical proofs and various proof techniques. The onset of this phase typically occurs during a 'transition to higher mathematics' course, an 'introduction to proofs' course or an introductory course in mathematical analysis or abstract algebra. Students of computer science or applied mathematics may encounter this formalism during their underclassman undergraduate years via a course in discrete mathematics. Exposure to counterexamples is typical, and an ability to discern between sound mathematical argumentation and erroneous mathematical argumentation is developed. This is an important phase for extinguishing misleading intuitions and elevating accurate intuitions, which the mathematical formalism aids with. The formality helps to deal with technical details, while intuition helps to deal with the bigger picture. However, sometimes a student can get stalled at this intermediary stage. This occurs when a student discards too much good intuition, rendering them able to only process mathematics at the formal level, instead of the more intuitive informal level. Getting stalled in this manner can impact the student's ability to read mathematical papers. | Upperclassman undergraduate years through early graduate years | Susceptibility to formal errors persists as a result of inapt formalism or an inability to perform "sanity checks" against intuition or other menial rules of thumb, such as a trivial sign error or the failure to verify an assumption made along the argumentation. However, unlike the previous stage such errors can usually be detected and often repaired once discovered. | The reintroduction of calculus in the context of rigorous ε, δ notation |
| 3 | Advanced | Post-rigorous | At this phase, students return to a more intuitive mindset, similar to that of the initial pre-rigorous phase, except now, misleading intuition has been extinguished, enabling students to reliably reason at a more informal level. The assimilation of mathematical formalism equips students with an instinctive feel for mathematical soundness. This allows them to ascertain an intuitive sketch of a mathematical argument that can subsequently be converted into a formal proof. The emphasis is now on applications, intuition, and the “big picture". This optimizes problem-solving efficiency because now formal rigor is infused with a refined intuition. In this ideal state, every heuristic argument naturally suggests its rigorous counterpart, and vice versa. This readies the practitioner to tackle complex mathematical problems that otherwise would be inaccessible. Students generally attain this phase towards their later graduate years, when they have typically begun reading mathematical research papers. | Later graduate years and beyond | Susceptibility to formal errors persists, albeit for a different reason; formalism is suspended for the sake of expedient intuitive reasoning and then later recalled for the translation into a rigorous argument, which can be sometimes be imprecise. | The improvised and provisional use of infinitesimals or big-O notation; temporarily constructing informal, quasi-rigorous argumentation that can be later converted into a formal, rigorous argumentation |